I thought I would go ahead and follow up with what I found, for future reference.
We did a lot of beach basking and bike riding, so I didn't explore the area too incredibly much, but what I found is that, if you are staying anywhere on the main Galveston island and you want to actually pull on to the beach with your vehicle and park there where you are chilling, aside from what Texas_WJ said, you have two main options that I found.
A) East beach
... the eastern tip of the island. Popular beach for partying, seeing as it is the only beach I saw where it is actually posted that alcohol is allowed on it. They also have a beer and concession stand.
B) Don't know the name of the beach, but if you cruise Seawall Blvd going west, you'll go past 61st street and keep an eye on the left for a bar called Woody's
. Right after that bar (not the actual turn in for the bar, but a few hundred feet down) is another beach entrance and it is posted there as well that you can park on the beach.
Both places post a price of $8 to park, but we were there Monday - Wednesday and went during day and night and never had anyone in either shack where you are supposed to pay, so of course, we didn't.
And, yah, I didn't play too much in the sand due to my street-tread tires and lack the fact that I didn't have anyone else to help with a pull in case I actually got the WJ somewhere quadra-drive wouldn't get out of.